MCQ
If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is
- A$792$
- B$252$
- ✓$462$
- D$330$
$N _{ C _{ r -1}}: N _{ C _{ r }}: N _{ C _{ r +1}}=5: 10: 14$
$\Rightarrow \frac{ N _{ C _{r}}}{ N _{ C _{ r -1}}}=\frac{ N +1- r }{ r }=2$
$\frac{N_{C_{r+1}}}{N_{C_{r}}}=\frac{N-r}{r+1}=\frac{7}{5}$
$\Rightarrow \quad r=4, N=11$
$\Rightarrow \quad(1+x)^{11}$
Largest coefficient $={ }^{11} C _{6}=462$
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$B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$
$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ .
If set $C$ is singleton set then sum of all possible values of $m$ is